THOMAS METTLER

We introduce a new functional \(\mathcal{E}_{\mathfrak{p}}\) on the space of conformal structures on an oriented projective manifold \((M,\mathfrak{p})\). The non-negative quantity \(\mathcal{E}_{\mathfrak{p}}([g])\) measures how much \(\mathfrak{p}\) deviates from being defined by a \([g]\)-conformal connection. In the case of a projective surface \((\Sigma,\mathfrak{p})\), we canonically construct an indefinite Kähler--Einstein structure \((h_{\mathfrak{p}},\Omega_{\mathfrak{p}})\) on the total space \(Y\) of a fibre bundle over \(\Sigma\) and show that a conformal structure \([g]\) is a critical point for \(\mathcal{E}_{\mathfrak{p}}\) if and only if a certain lift \(\widetilde{[g]} : (\Sigma,[g]) \to (Y,h_{\mathfrak{p}})\) is weakly conformal. In fact, in the compact case \(\mathcal{E}_{\mathfrak{p}}([g])\) is -- up to a topological constant -- just the Dirichlet energy of \(\widetilde{[g]}\). As an application, we prove a novel characterisation of properly convex projective structures among all flat projective structures. As a by-product, we obtain a Gauss--Bonnet type identity for oriented projective surfaces.

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We show that a properly convex projective structure \(\mathfrak{p}\) on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if \(\mathfrak{p}\) is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that \(\mathfrak{p}\) admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable \(L^2\)-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

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We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every properly convex projective surface arises from a unique minimal Lagrangian connection.

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